Nonlinear differential equations exhibit rich phenomena in many fields but are notoriously challenging to solve. Recently, Liu et al. [1] demonstrated the first efficient quantum algorithm for dissipative quadratic differential equations under the condition $R < 1$, where $R$ measures the ratio of nonlinearity to dissipation using the $\ell_2$ norm. Here we develop an efficient quantum algorithm based on [1] for reaction-diffusion equations, a class of nonlinear partial differential equations (PDEs). To achieve this, we improve upon the Carleman linearization approach introduced in [1] to obtain a faster convergence rate under the condition $R_D < 1$, where $R_D$ measures the ratio of nonlinearity to dissipation using the $\ell_{\infty}$ norm. Since $R_D$ is independent of the number of spatial grid points $n$ while $R$ increases with $n$, the criterion $R_D<1$ is significantly milder than $R<1$ for high-dimensional systems and can stay convergent under grid refinement for approximating PDEs. As applications of our quantum algorithm we consider the Fisher-KPP and Allen-Cahn equations, which have interpretations in classical physics. In particular, we show how to estimate the mean square kinetic energy in the solution by postprocessing the quantum state that encodes it to extract derivative information.

翻译：非线性微分方程在许多领域展现出丰富的现象，但求解极具挑战性。近期，Liu等人[1]在条件$R<1$下首次证明了耗散二次微分方程存在高效量子算法，其中$R$通过$\ell_2$范数衡量非线性项与耗散项的比值。本文基于[1]针对反应-扩散方程（一类非线性偏微分方程）提出了高效量子算法。为此，我们改进了[1]中引入的卡莱曼线性化方法，在条件$R_D<1$下实现了更快的收敛速度，其中$R_D$通过$\ell_{\infty}$范数衡量非线性项与耗散项的比值。由于$R_D$与空间网格点数$n$无关，而$R$随$n$增加而增大，因此对于高维系统，$R_D<1$的准则比$R<1$显著更宽松，且能在偏微分方程逼近的网格细化过程中保持收敛性。作为量子算法的应用示例，我们研究了具有经典物理解的Fisher-KPP方程和Allen-Cahn方程。特别地，我们展示了如何通过对编码解信息的量子态进行后处理来提取导数信息，从而估计解中的均方动能。